Daniel Janner QC, whose father was the late Labour peer Lord Janner, is the founder and secretary of Fair.
The group strongly believes in anonymity up until charge, and is also calling for a change in the language of criminal proceedings from victims to complainants.
Mr Janner described Sir Cliff’s support as an “enormous boost to our important work”.
“It is a privilege and honour that Sir Cliff has added his support. We campaign to re-balance the scales of justice to protect those falsely accused,” he said.
“At present we are concentrating our efforts to change the law to provide for anonymity until charge for those accused of sexual offences.”
Other Fair supporters include broadcaster Paul Gambaccini who was arrested in October 2013 over a claim he sexually assaulted two teenage boys as part of Operation Yewtree, set up in the wake of the revelations about paedophile Jimmy Savile.
The 69-year-old, a regular fixture on the airwaves for decades, spent a year on bail before the case was dropped.
Former Tory MP Harvey Proctor had his home raided and was publicly named, and is also a supporter of the pressure group.
He was investigated as part of Scotland Yard’s doomed sex abuse probe Operation Midland that centred on claims boys were sexually abused by a number of public figures more than 30 years ago.